Properties

Label 8034.2.a.bc.1.2
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 0
Dimension 15
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.77963\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.77963 q^{5} -1.00000 q^{6} +2.13919 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.77963 q^{5} -1.00000 q^{6} +2.13919 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.77963 q^{10} +4.21496 q^{11} -1.00000 q^{12} -1.00000 q^{13} +2.13919 q^{14} +3.77963 q^{15} +1.00000 q^{16} +4.66431 q^{17} +1.00000 q^{18} +8.26158 q^{19} -3.77963 q^{20} -2.13919 q^{21} +4.21496 q^{22} +0.374182 q^{23} -1.00000 q^{24} +9.28562 q^{25} -1.00000 q^{26} -1.00000 q^{27} +2.13919 q^{28} +10.1926 q^{29} +3.77963 q^{30} -3.35956 q^{31} +1.00000 q^{32} -4.21496 q^{33} +4.66431 q^{34} -8.08535 q^{35} +1.00000 q^{36} -5.93324 q^{37} +8.26158 q^{38} +1.00000 q^{39} -3.77963 q^{40} -10.2607 q^{41} -2.13919 q^{42} -7.02409 q^{43} +4.21496 q^{44} -3.77963 q^{45} +0.374182 q^{46} +5.37045 q^{47} -1.00000 q^{48} -2.42386 q^{49} +9.28562 q^{50} -4.66431 q^{51} -1.00000 q^{52} -3.53412 q^{53} -1.00000 q^{54} -15.9310 q^{55} +2.13919 q^{56} -8.26158 q^{57} +10.1926 q^{58} +5.68060 q^{59} +3.77963 q^{60} -5.36119 q^{61} -3.35956 q^{62} +2.13919 q^{63} +1.00000 q^{64} +3.77963 q^{65} -4.21496 q^{66} +8.01434 q^{67} +4.66431 q^{68} -0.374182 q^{69} -8.08535 q^{70} -8.03750 q^{71} +1.00000 q^{72} +12.5833 q^{73} -5.93324 q^{74} -9.28562 q^{75} +8.26158 q^{76} +9.01660 q^{77} +1.00000 q^{78} +13.2800 q^{79} -3.77963 q^{80} +1.00000 q^{81} -10.2607 q^{82} +4.49347 q^{83} -2.13919 q^{84} -17.6294 q^{85} -7.02409 q^{86} -10.1926 q^{87} +4.21496 q^{88} -11.6173 q^{89} -3.77963 q^{90} -2.13919 q^{91} +0.374182 q^{92} +3.35956 q^{93} +5.37045 q^{94} -31.2257 q^{95} -1.00000 q^{96} +10.7188 q^{97} -2.42386 q^{98} +4.21496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −3.77963 −1.69030 −0.845151 0.534527i \(-0.820490\pi\)
−0.845151 + 0.534527i \(0.820490\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.13919 0.808538 0.404269 0.914640i \(-0.367526\pi\)
0.404269 + 0.914640i \(0.367526\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.77963 −1.19522
\(11\) 4.21496 1.27086 0.635429 0.772159i \(-0.280823\pi\)
0.635429 + 0.772159i \(0.280823\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) 2.13919 0.571723
\(15\) 3.77963 0.975897
\(16\) 1.00000 0.250000
\(17\) 4.66431 1.13126 0.565631 0.824658i \(-0.308633\pi\)
0.565631 + 0.824658i \(0.308633\pi\)
\(18\) 1.00000 0.235702
\(19\) 8.26158 1.89534 0.947668 0.319257i \(-0.103433\pi\)
0.947668 + 0.319257i \(0.103433\pi\)
\(20\) −3.77963 −0.845151
\(21\) −2.13919 −0.466810
\(22\) 4.21496 0.898633
\(23\) 0.374182 0.0780223 0.0390112 0.999239i \(-0.487579\pi\)
0.0390112 + 0.999239i \(0.487579\pi\)
\(24\) −1.00000 −0.204124
\(25\) 9.28562 1.85712
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 2.13919 0.404269
\(29\) 10.1926 1.89271 0.946357 0.323123i \(-0.104733\pi\)
0.946357 + 0.323123i \(0.104733\pi\)
\(30\) 3.77963 0.690063
\(31\) −3.35956 −0.603395 −0.301697 0.953404i \(-0.597553\pi\)
−0.301697 + 0.953404i \(0.597553\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.21496 −0.733730
\(34\) 4.66431 0.799923
\(35\) −8.08535 −1.36667
\(36\) 1.00000 0.166667
\(37\) −5.93324 −0.975419 −0.487710 0.873006i \(-0.662168\pi\)
−0.487710 + 0.873006i \(0.662168\pi\)
\(38\) 8.26158 1.34021
\(39\) 1.00000 0.160128
\(40\) −3.77963 −0.597612
\(41\) −10.2607 −1.60245 −0.801224 0.598364i \(-0.795818\pi\)
−0.801224 + 0.598364i \(0.795818\pi\)
\(42\) −2.13919 −0.330084
\(43\) −7.02409 −1.07116 −0.535582 0.844483i \(-0.679908\pi\)
−0.535582 + 0.844483i \(0.679908\pi\)
\(44\) 4.21496 0.635429
\(45\) −3.77963 −0.563434
\(46\) 0.374182 0.0551701
\(47\) 5.37045 0.783361 0.391681 0.920101i \(-0.371894\pi\)
0.391681 + 0.920101i \(0.371894\pi\)
\(48\) −1.00000 −0.144338
\(49\) −2.42386 −0.346266
\(50\) 9.28562 1.31318
\(51\) −4.66431 −0.653135
\(52\) −1.00000 −0.138675
\(53\) −3.53412 −0.485449 −0.242724 0.970095i \(-0.578041\pi\)
−0.242724 + 0.970095i \(0.578041\pi\)
\(54\) −1.00000 −0.136083
\(55\) −15.9310 −2.14814
\(56\) 2.13919 0.285861
\(57\) −8.26158 −1.09427
\(58\) 10.1926 1.33835
\(59\) 5.68060 0.739551 0.369776 0.929121i \(-0.379435\pi\)
0.369776 + 0.929121i \(0.379435\pi\)
\(60\) 3.77963 0.487948
\(61\) −5.36119 −0.686430 −0.343215 0.939257i \(-0.611516\pi\)
−0.343215 + 0.939257i \(0.611516\pi\)
\(62\) −3.35956 −0.426665
\(63\) 2.13919 0.269513
\(64\) 1.00000 0.125000
\(65\) 3.77963 0.468806
\(66\) −4.21496 −0.518826
\(67\) 8.01434 0.979107 0.489553 0.871973i \(-0.337160\pi\)
0.489553 + 0.871973i \(0.337160\pi\)
\(68\) 4.66431 0.565631
\(69\) −0.374182 −0.0450462
\(70\) −8.08535 −0.966384
\(71\) −8.03750 −0.953876 −0.476938 0.878937i \(-0.658253\pi\)
−0.476938 + 0.878937i \(0.658253\pi\)
\(72\) 1.00000 0.117851
\(73\) 12.5833 1.47277 0.736383 0.676565i \(-0.236532\pi\)
0.736383 + 0.676565i \(0.236532\pi\)
\(74\) −5.93324 −0.689726
\(75\) −9.28562 −1.07221
\(76\) 8.26158 0.947668
\(77\) 9.01660 1.02754
\(78\) 1.00000 0.113228
\(79\) 13.2800 1.49411 0.747057 0.664759i \(-0.231466\pi\)
0.747057 + 0.664759i \(0.231466\pi\)
\(80\) −3.77963 −0.422576
\(81\) 1.00000 0.111111
\(82\) −10.2607 −1.13310
\(83\) 4.49347 0.493222 0.246611 0.969115i \(-0.420683\pi\)
0.246611 + 0.969115i \(0.420683\pi\)
\(84\) −2.13919 −0.233405
\(85\) −17.6294 −1.91218
\(86\) −7.02409 −0.757427
\(87\) −10.1926 −1.09276
\(88\) 4.21496 0.449316
\(89\) −11.6173 −1.23143 −0.615713 0.787970i \(-0.711132\pi\)
−0.615713 + 0.787970i \(0.711132\pi\)
\(90\) −3.77963 −0.398408
\(91\) −2.13919 −0.224248
\(92\) 0.374182 0.0390112
\(93\) 3.35956 0.348370
\(94\) 5.37045 0.553920
\(95\) −31.2257 −3.20369
\(96\) −1.00000 −0.102062
\(97\) 10.7188 1.08833 0.544166 0.838978i \(-0.316846\pi\)
0.544166 + 0.838978i \(0.316846\pi\)
\(98\) −2.42386 −0.244847
\(99\) 4.21496 0.423619
\(100\) 9.28562 0.928562
\(101\) −8.30659 −0.826536 −0.413268 0.910609i \(-0.635613\pi\)
−0.413268 + 0.910609i \(0.635613\pi\)
\(102\) −4.66431 −0.461836
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 8.08535 0.789050
\(106\) −3.53412 −0.343264
\(107\) −2.89268 −0.279646 −0.139823 0.990177i \(-0.544653\pi\)
−0.139823 + 0.990177i \(0.544653\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −12.7443 −1.22068 −0.610340 0.792139i \(-0.708967\pi\)
−0.610340 + 0.792139i \(0.708967\pi\)
\(110\) −15.9310 −1.51896
\(111\) 5.93324 0.563159
\(112\) 2.13919 0.202134
\(113\) 3.76628 0.354302 0.177151 0.984184i \(-0.443312\pi\)
0.177151 + 0.984184i \(0.443312\pi\)
\(114\) −8.26158 −0.773768
\(115\) −1.41427 −0.131881
\(116\) 10.1926 0.946357
\(117\) −1.00000 −0.0924500
\(118\) 5.68060 0.522942
\(119\) 9.97786 0.914669
\(120\) 3.77963 0.345032
\(121\) 6.76589 0.615081
\(122\) −5.36119 −0.485379
\(123\) 10.2607 0.925174
\(124\) −3.35956 −0.301697
\(125\) −16.1981 −1.44880
\(126\) 2.13919 0.190574
\(127\) −0.794271 −0.0704801 −0.0352401 0.999379i \(-0.511220\pi\)
−0.0352401 + 0.999379i \(0.511220\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.02409 0.618436
\(130\) 3.77963 0.331496
\(131\) −14.1470 −1.23603 −0.618013 0.786168i \(-0.712062\pi\)
−0.618013 + 0.786168i \(0.712062\pi\)
\(132\) −4.21496 −0.366865
\(133\) 17.6731 1.53245
\(134\) 8.01434 0.692333
\(135\) 3.77963 0.325299
\(136\) 4.66431 0.399962
\(137\) −11.3093 −0.966216 −0.483108 0.875561i \(-0.660492\pi\)
−0.483108 + 0.875561i \(0.660492\pi\)
\(138\) −0.374182 −0.0318525
\(139\) 20.2607 1.71849 0.859247 0.511562i \(-0.170933\pi\)
0.859247 + 0.511562i \(0.170933\pi\)
\(140\) −8.08535 −0.683337
\(141\) −5.37045 −0.452274
\(142\) −8.03750 −0.674492
\(143\) −4.21496 −0.352473
\(144\) 1.00000 0.0833333
\(145\) −38.5242 −3.19926
\(146\) 12.5833 1.04140
\(147\) 2.42386 0.199917
\(148\) −5.93324 −0.487710
\(149\) 19.6482 1.60964 0.804820 0.593518i \(-0.202262\pi\)
0.804820 + 0.593518i \(0.202262\pi\)
\(150\) −9.28562 −0.758168
\(151\) 6.34285 0.516174 0.258087 0.966122i \(-0.416908\pi\)
0.258087 + 0.966122i \(0.416908\pi\)
\(152\) 8.26158 0.670103
\(153\) 4.66431 0.377087
\(154\) 9.01660 0.726578
\(155\) 12.6979 1.01992
\(156\) 1.00000 0.0800641
\(157\) 9.90015 0.790118 0.395059 0.918656i \(-0.370724\pi\)
0.395059 + 0.918656i \(0.370724\pi\)
\(158\) 13.2800 1.05650
\(159\) 3.53412 0.280274
\(160\) −3.77963 −0.298806
\(161\) 0.800447 0.0630840
\(162\) 1.00000 0.0785674
\(163\) 13.8077 1.08150 0.540751 0.841182i \(-0.318140\pi\)
0.540751 + 0.841182i \(0.318140\pi\)
\(164\) −10.2607 −0.801224
\(165\) 15.9310 1.24023
\(166\) 4.49347 0.348761
\(167\) 1.13315 0.0876854 0.0438427 0.999038i \(-0.486040\pi\)
0.0438427 + 0.999038i \(0.486040\pi\)
\(168\) −2.13919 −0.165042
\(169\) 1.00000 0.0769231
\(170\) −17.6294 −1.35211
\(171\) 8.26158 0.631779
\(172\) −7.02409 −0.535582
\(173\) −14.1248 −1.07389 −0.536945 0.843617i \(-0.680422\pi\)
−0.536945 + 0.843617i \(0.680422\pi\)
\(174\) −10.1926 −0.772697
\(175\) 19.8637 1.50156
\(176\) 4.21496 0.317715
\(177\) −5.68060 −0.426980
\(178\) −11.6173 −0.870750
\(179\) 18.7283 1.39982 0.699908 0.714233i \(-0.253224\pi\)
0.699908 + 0.714233i \(0.253224\pi\)
\(180\) −3.77963 −0.281717
\(181\) 7.86910 0.584905 0.292453 0.956280i \(-0.405529\pi\)
0.292453 + 0.956280i \(0.405529\pi\)
\(182\) −2.13919 −0.158567
\(183\) 5.36119 0.396311
\(184\) 0.374182 0.0275851
\(185\) 22.4255 1.64875
\(186\) 3.35956 0.246335
\(187\) 19.6599 1.43767
\(188\) 5.37045 0.391681
\(189\) −2.13919 −0.155603
\(190\) −31.2257 −2.26535
\(191\) 5.36373 0.388106 0.194053 0.980991i \(-0.437837\pi\)
0.194053 + 0.980991i \(0.437837\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 20.9799 1.51017 0.755085 0.655627i \(-0.227596\pi\)
0.755085 + 0.655627i \(0.227596\pi\)
\(194\) 10.7188 0.769567
\(195\) −3.77963 −0.270665
\(196\) −2.42386 −0.173133
\(197\) 5.11889 0.364706 0.182353 0.983233i \(-0.441629\pi\)
0.182353 + 0.983233i \(0.441629\pi\)
\(198\) 4.21496 0.299544
\(199\) 21.6419 1.53415 0.767076 0.641556i \(-0.221711\pi\)
0.767076 + 0.641556i \(0.221711\pi\)
\(200\) 9.28562 0.656592
\(201\) −8.01434 −0.565288
\(202\) −8.30659 −0.584449
\(203\) 21.8039 1.53033
\(204\) −4.66431 −0.326567
\(205\) 38.7816 2.70862
\(206\) −1.00000 −0.0696733
\(207\) 0.374182 0.0260074
\(208\) −1.00000 −0.0693375
\(209\) 34.8222 2.40870
\(210\) 8.08535 0.557942
\(211\) 20.6224 1.41971 0.709853 0.704350i \(-0.248762\pi\)
0.709853 + 0.704350i \(0.248762\pi\)
\(212\) −3.53412 −0.242724
\(213\) 8.03750 0.550721
\(214\) −2.89268 −0.197739
\(215\) 26.5485 1.81059
\(216\) −1.00000 −0.0680414
\(217\) −7.18674 −0.487868
\(218\) −12.7443 −0.863152
\(219\) −12.5833 −0.850302
\(220\) −15.9310 −1.07407
\(221\) −4.66431 −0.313756
\(222\) 5.93324 0.398213
\(223\) −6.51667 −0.436389 −0.218194 0.975905i \(-0.570017\pi\)
−0.218194 + 0.975905i \(0.570017\pi\)
\(224\) 2.13919 0.142931
\(225\) 9.28562 0.619041
\(226\) 3.76628 0.250529
\(227\) 0.198682 0.0131870 0.00659348 0.999978i \(-0.497901\pi\)
0.00659348 + 0.999978i \(0.497901\pi\)
\(228\) −8.26158 −0.547136
\(229\) −26.2874 −1.73712 −0.868559 0.495586i \(-0.834954\pi\)
−0.868559 + 0.495586i \(0.834954\pi\)
\(230\) −1.41427 −0.0932542
\(231\) −9.01660 −0.593249
\(232\) 10.1926 0.669175
\(233\) −26.4007 −1.72957 −0.864784 0.502145i \(-0.832545\pi\)
−0.864784 + 0.502145i \(0.832545\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −20.2983 −1.32412
\(236\) 5.68060 0.369776
\(237\) −13.2800 −0.862628
\(238\) 9.97786 0.646768
\(239\) −1.10359 −0.0713853 −0.0356927 0.999363i \(-0.511364\pi\)
−0.0356927 + 0.999363i \(0.511364\pi\)
\(240\) 3.77963 0.243974
\(241\) 0.969052 0.0624221 0.0312111 0.999513i \(-0.490064\pi\)
0.0312111 + 0.999513i \(0.490064\pi\)
\(242\) 6.76589 0.434928
\(243\) −1.00000 −0.0641500
\(244\) −5.36119 −0.343215
\(245\) 9.16132 0.585295
\(246\) 10.2607 0.654197
\(247\) −8.26158 −0.525672
\(248\) −3.35956 −0.213332
\(249\) −4.49347 −0.284762
\(250\) −16.1981 −1.02446
\(251\) −14.0977 −0.889836 −0.444918 0.895571i \(-0.646767\pi\)
−0.444918 + 0.895571i \(0.646767\pi\)
\(252\) 2.13919 0.134756
\(253\) 1.57716 0.0991553
\(254\) −0.794271 −0.0498370
\(255\) 17.6294 1.10400
\(256\) 1.00000 0.0625000
\(257\) −27.4037 −1.70940 −0.854698 0.519126i \(-0.826258\pi\)
−0.854698 + 0.519126i \(0.826258\pi\)
\(258\) 7.02409 0.437301
\(259\) −12.6923 −0.788664
\(260\) 3.77963 0.234403
\(261\) 10.1926 0.630905
\(262\) −14.1470 −0.874003
\(263\) 19.1820 1.18281 0.591407 0.806373i \(-0.298573\pi\)
0.591407 + 0.806373i \(0.298573\pi\)
\(264\) −4.21496 −0.259413
\(265\) 13.3577 0.820556
\(266\) 17.6731 1.08361
\(267\) 11.6173 0.710964
\(268\) 8.01434 0.489553
\(269\) −19.1913 −1.17012 −0.585058 0.810991i \(-0.698928\pi\)
−0.585058 + 0.810991i \(0.698928\pi\)
\(270\) 3.77963 0.230021
\(271\) −21.9819 −1.33531 −0.667653 0.744473i \(-0.732701\pi\)
−0.667653 + 0.744473i \(0.732701\pi\)
\(272\) 4.66431 0.282816
\(273\) 2.13919 0.129470
\(274\) −11.3093 −0.683218
\(275\) 39.1385 2.36014
\(276\) −0.374182 −0.0225231
\(277\) 14.0129 0.841956 0.420978 0.907071i \(-0.361687\pi\)
0.420978 + 0.907071i \(0.361687\pi\)
\(278\) 20.2607 1.21516
\(279\) −3.35956 −0.201132
\(280\) −8.08535 −0.483192
\(281\) −6.84656 −0.408432 −0.204216 0.978926i \(-0.565464\pi\)
−0.204216 + 0.978926i \(0.565464\pi\)
\(282\) −5.37045 −0.319806
\(283\) 22.2633 1.32342 0.661708 0.749762i \(-0.269832\pi\)
0.661708 + 0.749762i \(0.269832\pi\)
\(284\) −8.03750 −0.476938
\(285\) 31.2257 1.84965
\(286\) −4.21496 −0.249236
\(287\) −21.9495 −1.29564
\(288\) 1.00000 0.0589256
\(289\) 4.75583 0.279755
\(290\) −38.5242 −2.26222
\(291\) −10.7188 −0.628349
\(292\) 12.5833 0.736383
\(293\) 0.00371315 0.000216925 0 0.000108462 1.00000i \(-0.499965\pi\)
0.000108462 1.00000i \(0.499965\pi\)
\(294\) 2.42386 0.141363
\(295\) −21.4706 −1.25007
\(296\) −5.93324 −0.344863
\(297\) −4.21496 −0.244577
\(298\) 19.6482 1.13819
\(299\) −0.374182 −0.0216395
\(300\) −9.28562 −0.536105
\(301\) −15.0259 −0.866076
\(302\) 6.34285 0.364990
\(303\) 8.30659 0.477201
\(304\) 8.26158 0.473834
\(305\) 20.2633 1.16028
\(306\) 4.66431 0.266641
\(307\) 11.9442 0.681692 0.340846 0.940119i \(-0.389287\pi\)
0.340846 + 0.940119i \(0.389287\pi\)
\(308\) 9.01660 0.513769
\(309\) 1.00000 0.0568880
\(310\) 12.6979 0.721192
\(311\) −4.77869 −0.270975 −0.135487 0.990779i \(-0.543260\pi\)
−0.135487 + 0.990779i \(0.543260\pi\)
\(312\) 1.00000 0.0566139
\(313\) −17.6126 −0.995525 −0.497762 0.867313i \(-0.665845\pi\)
−0.497762 + 0.867313i \(0.665845\pi\)
\(314\) 9.90015 0.558698
\(315\) −8.08535 −0.455558
\(316\) 13.2800 0.747057
\(317\) −17.4337 −0.979176 −0.489588 0.871954i \(-0.662853\pi\)
−0.489588 + 0.871954i \(0.662853\pi\)
\(318\) 3.53412 0.198184
\(319\) 42.9613 2.40537
\(320\) −3.77963 −0.211288
\(321\) 2.89268 0.161453
\(322\) 0.800447 0.0446071
\(323\) 38.5346 2.14412
\(324\) 1.00000 0.0555556
\(325\) −9.28562 −0.515073
\(326\) 13.8077 0.764738
\(327\) 12.7443 0.704760
\(328\) −10.2607 −0.566551
\(329\) 11.4884 0.633377
\(330\) 15.9310 0.876973
\(331\) 11.6141 0.638369 0.319184 0.947693i \(-0.396591\pi\)
0.319184 + 0.947693i \(0.396591\pi\)
\(332\) 4.49347 0.246611
\(333\) −5.93324 −0.325140
\(334\) 1.13315 0.0620029
\(335\) −30.2912 −1.65499
\(336\) −2.13919 −0.116702
\(337\) −17.6276 −0.960237 −0.480119 0.877204i \(-0.659406\pi\)
−0.480119 + 0.877204i \(0.659406\pi\)
\(338\) 1.00000 0.0543928
\(339\) −3.76628 −0.204556
\(340\) −17.6294 −0.956088
\(341\) −14.1604 −0.766829
\(342\) 8.26158 0.446735
\(343\) −20.1594 −1.08851
\(344\) −7.02409 −0.378713
\(345\) 1.41427 0.0761418
\(346\) −14.1248 −0.759355
\(347\) −3.93905 −0.211459 −0.105730 0.994395i \(-0.533718\pi\)
−0.105730 + 0.994395i \(0.533718\pi\)
\(348\) −10.1926 −0.546379
\(349\) 26.4315 1.41484 0.707422 0.706791i \(-0.249858\pi\)
0.707422 + 0.706791i \(0.249858\pi\)
\(350\) 19.8637 1.06176
\(351\) 1.00000 0.0533761
\(352\) 4.21496 0.224658
\(353\) 8.95497 0.476625 0.238312 0.971189i \(-0.423406\pi\)
0.238312 + 0.971189i \(0.423406\pi\)
\(354\) −5.68060 −0.301920
\(355\) 30.3788 1.61234
\(356\) −11.6173 −0.615713
\(357\) −9.97786 −0.528084
\(358\) 18.7283 0.989819
\(359\) 4.61075 0.243346 0.121673 0.992570i \(-0.461174\pi\)
0.121673 + 0.992570i \(0.461174\pi\)
\(360\) −3.77963 −0.199204
\(361\) 49.2537 2.59230
\(362\) 7.86910 0.413591
\(363\) −6.76589 −0.355117
\(364\) −2.13919 −0.112124
\(365\) −47.5603 −2.48942
\(366\) 5.36119 0.280234
\(367\) 8.74574 0.456524 0.228262 0.973600i \(-0.426696\pi\)
0.228262 + 0.973600i \(0.426696\pi\)
\(368\) 0.374182 0.0195056
\(369\) −10.2607 −0.534149
\(370\) 22.4255 1.16585
\(371\) −7.56016 −0.392504
\(372\) 3.35956 0.174185
\(373\) 17.1101 0.885929 0.442965 0.896539i \(-0.353927\pi\)
0.442965 + 0.896539i \(0.353927\pi\)
\(374\) 19.6599 1.01659
\(375\) 16.1981 0.836464
\(376\) 5.37045 0.276960
\(377\) −10.1926 −0.524944
\(378\) −2.13919 −0.110028
\(379\) 14.7409 0.757187 0.378593 0.925563i \(-0.376408\pi\)
0.378593 + 0.925563i \(0.376408\pi\)
\(380\) −31.2257 −1.60185
\(381\) 0.794271 0.0406917
\(382\) 5.36373 0.274432
\(383\) −3.02320 −0.154478 −0.0772392 0.997013i \(-0.524611\pi\)
−0.0772392 + 0.997013i \(0.524611\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −34.0794 −1.73685
\(386\) 20.9799 1.06785
\(387\) −7.02409 −0.357054
\(388\) 10.7188 0.544166
\(389\) −28.4470 −1.44232 −0.721161 0.692768i \(-0.756391\pi\)
−0.721161 + 0.692768i \(0.756391\pi\)
\(390\) −3.77963 −0.191389
\(391\) 1.74530 0.0882637
\(392\) −2.42386 −0.122424
\(393\) 14.1470 0.713620
\(394\) 5.11889 0.257886
\(395\) −50.1935 −2.52551
\(396\) 4.21496 0.211810
\(397\) 21.7150 1.08985 0.544923 0.838486i \(-0.316559\pi\)
0.544923 + 0.838486i \(0.316559\pi\)
\(398\) 21.6419 1.08481
\(399\) −17.6731 −0.884761
\(400\) 9.28562 0.464281
\(401\) 15.6403 0.781039 0.390520 0.920595i \(-0.372295\pi\)
0.390520 + 0.920595i \(0.372295\pi\)
\(402\) −8.01434 −0.399719
\(403\) 3.35956 0.167352
\(404\) −8.30659 −0.413268
\(405\) −3.77963 −0.187811
\(406\) 21.8039 1.08211
\(407\) −25.0084 −1.23962
\(408\) −4.66431 −0.230918
\(409\) 11.2738 0.557451 0.278726 0.960371i \(-0.410088\pi\)
0.278726 + 0.960371i \(0.410088\pi\)
\(410\) 38.7816 1.91529
\(411\) 11.3093 0.557845
\(412\) −1.00000 −0.0492665
\(413\) 12.1519 0.597955
\(414\) 0.374182 0.0183900
\(415\) −16.9837 −0.833695
\(416\) −1.00000 −0.0490290
\(417\) −20.2607 −0.992172
\(418\) 34.8222 1.70321
\(419\) 6.51719 0.318386 0.159193 0.987248i \(-0.449111\pi\)
0.159193 + 0.987248i \(0.449111\pi\)
\(420\) 8.08535 0.394525
\(421\) 18.6962 0.911197 0.455599 0.890185i \(-0.349425\pi\)
0.455599 + 0.890185i \(0.349425\pi\)
\(422\) 20.6224 1.00388
\(423\) 5.37045 0.261120
\(424\) −3.53412 −0.171632
\(425\) 43.3110 2.10089
\(426\) 8.03750 0.389418
\(427\) −11.4686 −0.555005
\(428\) −2.89268 −0.139823
\(429\) 4.21496 0.203500
\(430\) 26.5485 1.28028
\(431\) −18.5036 −0.891287 −0.445644 0.895210i \(-0.647025\pi\)
−0.445644 + 0.895210i \(0.647025\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −29.7200 −1.42825 −0.714126 0.700017i \(-0.753176\pi\)
−0.714126 + 0.700017i \(0.753176\pi\)
\(434\) −7.18674 −0.344974
\(435\) 38.5242 1.84709
\(436\) −12.7443 −0.610340
\(437\) 3.09133 0.147879
\(438\) −12.5833 −0.601254
\(439\) −19.7906 −0.944555 −0.472278 0.881450i \(-0.656568\pi\)
−0.472278 + 0.881450i \(0.656568\pi\)
\(440\) −15.9310 −0.759481
\(441\) −2.42386 −0.115422
\(442\) −4.66431 −0.221859
\(443\) 7.33689 0.348586 0.174293 0.984694i \(-0.444236\pi\)
0.174293 + 0.984694i \(0.444236\pi\)
\(444\) 5.93324 0.281579
\(445\) 43.9090 2.08148
\(446\) −6.51667 −0.308573
\(447\) −19.6482 −0.929326
\(448\) 2.13919 0.101067
\(449\) 4.87953 0.230279 0.115140 0.993349i \(-0.463268\pi\)
0.115140 + 0.993349i \(0.463268\pi\)
\(450\) 9.28562 0.437728
\(451\) −43.2483 −2.03648
\(452\) 3.76628 0.177151
\(453\) −6.34285 −0.298013
\(454\) 0.198682 0.00932459
\(455\) 8.08535 0.379047
\(456\) −8.26158 −0.386884
\(457\) −20.0349 −0.937192 −0.468596 0.883413i \(-0.655240\pi\)
−0.468596 + 0.883413i \(0.655240\pi\)
\(458\) −26.2874 −1.22833
\(459\) −4.66431 −0.217712
\(460\) −1.41427 −0.0659407
\(461\) −18.6206 −0.867248 −0.433624 0.901094i \(-0.642765\pi\)
−0.433624 + 0.901094i \(0.642765\pi\)
\(462\) −9.01660 −0.419490
\(463\) 0.740569 0.0344172 0.0172086 0.999852i \(-0.494522\pi\)
0.0172086 + 0.999852i \(0.494522\pi\)
\(464\) 10.1926 0.473178
\(465\) −12.6979 −0.588851
\(466\) −26.4007 −1.22299
\(467\) −13.1137 −0.606830 −0.303415 0.952859i \(-0.598127\pi\)
−0.303415 + 0.952859i \(0.598127\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 17.1442 0.791645
\(470\) −20.2983 −0.936293
\(471\) −9.90015 −0.456175
\(472\) 5.68060 0.261471
\(473\) −29.6063 −1.36130
\(474\) −13.2800 −0.609970
\(475\) 76.7139 3.51987
\(476\) 9.97786 0.457334
\(477\) −3.53412 −0.161816
\(478\) −1.10359 −0.0504770
\(479\) 18.4924 0.844938 0.422469 0.906377i \(-0.361163\pi\)
0.422469 + 0.906377i \(0.361163\pi\)
\(480\) 3.77963 0.172516
\(481\) 5.93324 0.270533
\(482\) 0.969052 0.0441391
\(483\) −0.800447 −0.0364216
\(484\) 6.76589 0.307540
\(485\) −40.5132 −1.83961
\(486\) −1.00000 −0.0453609
\(487\) 32.5538 1.47515 0.737577 0.675263i \(-0.235970\pi\)
0.737577 + 0.675263i \(0.235970\pi\)
\(488\) −5.36119 −0.242690
\(489\) −13.8077 −0.624406
\(490\) 9.16132 0.413866
\(491\) 34.3094 1.54836 0.774181 0.632965i \(-0.218162\pi\)
0.774181 + 0.632965i \(0.218162\pi\)
\(492\) 10.2607 0.462587
\(493\) 47.5414 2.14116
\(494\) −8.26158 −0.371706
\(495\) −15.9310 −0.716045
\(496\) −3.35956 −0.150849
\(497\) −17.1937 −0.771245
\(498\) −4.49347 −0.201357
\(499\) −3.03217 −0.135739 −0.0678693 0.997694i \(-0.521620\pi\)
−0.0678693 + 0.997694i \(0.521620\pi\)
\(500\) −16.1981 −0.724399
\(501\) −1.13315 −0.0506252
\(502\) −14.0977 −0.629209
\(503\) −18.4993 −0.824843 −0.412421 0.910993i \(-0.635317\pi\)
−0.412421 + 0.910993i \(0.635317\pi\)
\(504\) 2.13919 0.0952871
\(505\) 31.3958 1.39710
\(506\) 1.57716 0.0701134
\(507\) −1.00000 −0.0444116
\(508\) −0.794271 −0.0352401
\(509\) −15.1859 −0.673102 −0.336551 0.941665i \(-0.609260\pi\)
−0.336551 + 0.941665i \(0.609260\pi\)
\(510\) 17.6294 0.780643
\(511\) 26.9181 1.19079
\(512\) 1.00000 0.0441942
\(513\) −8.26158 −0.364758
\(514\) −27.4037 −1.20873
\(515\) 3.77963 0.166550
\(516\) 7.02409 0.309218
\(517\) 22.6363 0.995541
\(518\) −12.6923 −0.557669
\(519\) 14.1248 0.620011
\(520\) 3.77963 0.165748
\(521\) 24.7979 1.08641 0.543207 0.839599i \(-0.317210\pi\)
0.543207 + 0.839599i \(0.317210\pi\)
\(522\) 10.1926 0.446117
\(523\) 10.9746 0.479886 0.239943 0.970787i \(-0.422871\pi\)
0.239943 + 0.970787i \(0.422871\pi\)
\(524\) −14.1470 −0.618013
\(525\) −19.8637 −0.866923
\(526\) 19.1820 0.836376
\(527\) −15.6700 −0.682598
\(528\) −4.21496 −0.183433
\(529\) −22.8600 −0.993913
\(530\) 13.3577 0.580220
\(531\) 5.68060 0.246517
\(532\) 17.6731 0.766226
\(533\) 10.2607 0.444439
\(534\) 11.6173 0.502728
\(535\) 10.9333 0.472686
\(536\) 8.01434 0.346167
\(537\) −18.7283 −0.808184
\(538\) −19.1913 −0.827397
\(539\) −10.2165 −0.440055
\(540\) 3.77963 0.162649
\(541\) 27.7843 1.19454 0.597271 0.802040i \(-0.296252\pi\)
0.597271 + 0.802040i \(0.296252\pi\)
\(542\) −21.9819 −0.944204
\(543\) −7.86910 −0.337695
\(544\) 4.66431 0.199981
\(545\) 48.1687 2.06332
\(546\) 2.13919 0.0915489
\(547\) −13.7113 −0.586255 −0.293127 0.956073i \(-0.594696\pi\)
−0.293127 + 0.956073i \(0.594696\pi\)
\(548\) −11.3093 −0.483108
\(549\) −5.36119 −0.228810
\(550\) 39.1385 1.66887
\(551\) 84.2068 3.58733
\(552\) −0.374182 −0.0159262
\(553\) 28.4084 1.20805
\(554\) 14.0129 0.595352
\(555\) −22.4255 −0.951909
\(556\) 20.2607 0.859247
\(557\) −12.1248 −0.513746 −0.256873 0.966445i \(-0.582692\pi\)
−0.256873 + 0.966445i \(0.582692\pi\)
\(558\) −3.35956 −0.142222
\(559\) 7.02409 0.297087
\(560\) −8.08535 −0.341669
\(561\) −19.6599 −0.830042
\(562\) −6.84656 −0.288805
\(563\) −19.4739 −0.820727 −0.410364 0.911922i \(-0.634598\pi\)
−0.410364 + 0.911922i \(0.634598\pi\)
\(564\) −5.37045 −0.226137
\(565\) −14.2351 −0.598877
\(566\) 22.2633 0.935796
\(567\) 2.13919 0.0898376
\(568\) −8.03750 −0.337246
\(569\) 36.7811 1.54194 0.770972 0.636869i \(-0.219771\pi\)
0.770972 + 0.636869i \(0.219771\pi\)
\(570\) 31.2257 1.30790
\(571\) 37.9425 1.58784 0.793921 0.608021i \(-0.208036\pi\)
0.793921 + 0.608021i \(0.208036\pi\)
\(572\) −4.21496 −0.176236
\(573\) −5.36373 −0.224073
\(574\) −21.9495 −0.916156
\(575\) 3.47451 0.144897
\(576\) 1.00000 0.0416667
\(577\) −38.2929 −1.59415 −0.797077 0.603878i \(-0.793621\pi\)
−0.797077 + 0.603878i \(0.793621\pi\)
\(578\) 4.75583 0.197816
\(579\) −20.9799 −0.871897
\(580\) −38.5242 −1.59963
\(581\) 9.61238 0.398789
\(582\) −10.7188 −0.444310
\(583\) −14.8962 −0.616937
\(584\) 12.5833 0.520701
\(585\) 3.77963 0.156269
\(586\) 0.00371315 0.000153389 0
\(587\) 12.8587 0.530735 0.265368 0.964147i \(-0.414507\pi\)
0.265368 + 0.964147i \(0.414507\pi\)
\(588\) 2.42386 0.0999585
\(589\) −27.7553 −1.14364
\(590\) −21.4706 −0.883930
\(591\) −5.11889 −0.210563
\(592\) −5.93324 −0.243855
\(593\) −34.5814 −1.42009 −0.710044 0.704157i \(-0.751325\pi\)
−0.710044 + 0.704157i \(0.751325\pi\)
\(594\) −4.21496 −0.172942
\(595\) −37.7126 −1.54607
\(596\) 19.6482 0.804820
\(597\) −21.6419 −0.885743
\(598\) −0.374182 −0.0153014
\(599\) 38.7515 1.58334 0.791672 0.610947i \(-0.209211\pi\)
0.791672 + 0.610947i \(0.209211\pi\)
\(600\) −9.28562 −0.379084
\(601\) −33.1927 −1.35396 −0.676980 0.736002i \(-0.736712\pi\)
−0.676980 + 0.736002i \(0.736712\pi\)
\(602\) −15.0259 −0.612408
\(603\) 8.01434 0.326369
\(604\) 6.34285 0.258087
\(605\) −25.5726 −1.03967
\(606\) 8.30659 0.337432
\(607\) 35.9102 1.45755 0.728774 0.684754i \(-0.240090\pi\)
0.728774 + 0.684754i \(0.240090\pi\)
\(608\) 8.26158 0.335051
\(609\) −21.8039 −0.883537
\(610\) 20.2633 0.820438
\(611\) −5.37045 −0.217265
\(612\) 4.66431 0.188544
\(613\) −10.8035 −0.436350 −0.218175 0.975910i \(-0.570010\pi\)
−0.218175 + 0.975910i \(0.570010\pi\)
\(614\) 11.9442 0.482029
\(615\) −38.7816 −1.56382
\(616\) 9.01660 0.363289
\(617\) −15.5335 −0.625353 −0.312677 0.949860i \(-0.601226\pi\)
−0.312677 + 0.949860i \(0.601226\pi\)
\(618\) 1.00000 0.0402259
\(619\) −2.27622 −0.0914892 −0.0457446 0.998953i \(-0.514566\pi\)
−0.0457446 + 0.998953i \(0.514566\pi\)
\(620\) 12.6979 0.509960
\(621\) −0.374182 −0.0150154
\(622\) −4.77869 −0.191608
\(623\) −24.8515 −0.995655
\(624\) 1.00000 0.0400320
\(625\) 14.7946 0.591785
\(626\) −17.6126 −0.703942
\(627\) −34.8222 −1.39067
\(628\) 9.90015 0.395059
\(629\) −27.6745 −1.10346
\(630\) −8.08535 −0.322128
\(631\) −15.6555 −0.623237 −0.311618 0.950207i \(-0.600871\pi\)
−0.311618 + 0.950207i \(0.600871\pi\)
\(632\) 13.2800 0.528249
\(633\) −20.6224 −0.819667
\(634\) −17.4337 −0.692382
\(635\) 3.00205 0.119133
\(636\) 3.53412 0.140137
\(637\) 2.42386 0.0960370
\(638\) 42.9613 1.70085
\(639\) −8.03750 −0.317959
\(640\) −3.77963 −0.149403
\(641\) −37.6748 −1.48806 −0.744032 0.668144i \(-0.767089\pi\)
−0.744032 + 0.668144i \(0.767089\pi\)
\(642\) 2.89268 0.114165
\(643\) 15.7057 0.619372 0.309686 0.950839i \(-0.399776\pi\)
0.309686 + 0.950839i \(0.399776\pi\)
\(644\) 0.800447 0.0315420
\(645\) −26.5485 −1.04534
\(646\) 38.5346 1.51612
\(647\) −41.1437 −1.61753 −0.808763 0.588135i \(-0.799862\pi\)
−0.808763 + 0.588135i \(0.799862\pi\)
\(648\) 1.00000 0.0392837
\(649\) 23.9435 0.939865
\(650\) −9.28562 −0.364212
\(651\) 7.18674 0.281670
\(652\) 13.8077 0.540751
\(653\) −22.8146 −0.892803 −0.446401 0.894833i \(-0.647295\pi\)
−0.446401 + 0.894833i \(0.647295\pi\)
\(654\) 12.7443 0.498341
\(655\) 53.4704 2.08926
\(656\) −10.2607 −0.400612
\(657\) 12.5833 0.490922
\(658\) 11.4884 0.447865
\(659\) 19.9616 0.777595 0.388797 0.921323i \(-0.372891\pi\)
0.388797 + 0.921323i \(0.372891\pi\)
\(660\) 15.9310 0.620113
\(661\) 11.2835 0.438878 0.219439 0.975626i \(-0.429577\pi\)
0.219439 + 0.975626i \(0.429577\pi\)
\(662\) 11.6141 0.451395
\(663\) 4.66431 0.181147
\(664\) 4.49347 0.174380
\(665\) −66.7978 −2.59031
\(666\) −5.93324 −0.229909
\(667\) 3.81388 0.147674
\(668\) 1.13315 0.0438427
\(669\) 6.51667 0.251949
\(670\) −30.2912 −1.17025
\(671\) −22.5972 −0.872356
\(672\) −2.13919 −0.0825211
\(673\) 6.29563 0.242679 0.121339 0.992611i \(-0.461281\pi\)
0.121339 + 0.992611i \(0.461281\pi\)
\(674\) −17.6276 −0.678990
\(675\) −9.28562 −0.357404
\(676\) 1.00000 0.0384615
\(677\) 7.71492 0.296509 0.148254 0.988949i \(-0.452635\pi\)
0.148254 + 0.988949i \(0.452635\pi\)
\(678\) −3.76628 −0.144643
\(679\) 22.9296 0.879958
\(680\) −17.6294 −0.676056
\(681\) −0.198682 −0.00761350
\(682\) −14.1604 −0.542230
\(683\) 51.5019 1.97066 0.985332 0.170646i \(-0.0545854\pi\)
0.985332 + 0.170646i \(0.0545854\pi\)
\(684\) 8.26158 0.315889
\(685\) 42.7449 1.63320
\(686\) −20.1594 −0.769691
\(687\) 26.2874 1.00293
\(688\) −7.02409 −0.267791
\(689\) 3.53412 0.134639
\(690\) 1.41427 0.0538404
\(691\) −28.5639 −1.08662 −0.543312 0.839531i \(-0.682830\pi\)
−0.543312 + 0.839531i \(0.682830\pi\)
\(692\) −14.1248 −0.536945
\(693\) 9.01660 0.342512
\(694\) −3.93905 −0.149524
\(695\) −76.5781 −2.90477
\(696\) −10.1926 −0.386349
\(697\) −47.8590 −1.81279
\(698\) 26.4315 1.00045
\(699\) 26.4007 0.998566
\(700\) 19.8637 0.750778
\(701\) 47.3302 1.78764 0.893819 0.448427i \(-0.148016\pi\)
0.893819 + 0.448427i \(0.148016\pi\)
\(702\) 1.00000 0.0377426
\(703\) −49.0180 −1.84875
\(704\) 4.21496 0.158857
\(705\) 20.2983 0.764480
\(706\) 8.95497 0.337025
\(707\) −17.7694 −0.668286
\(708\) −5.68060 −0.213490
\(709\) 22.4875 0.844537 0.422268 0.906471i \(-0.361234\pi\)
0.422268 + 0.906471i \(0.361234\pi\)
\(710\) 30.3788 1.14010
\(711\) 13.2800 0.498038
\(712\) −11.6173 −0.435375
\(713\) −1.25709 −0.0470783
\(714\) −9.97786 −0.373412
\(715\) 15.9310 0.595786
\(716\) 18.7283 0.699908
\(717\) 1.10359 0.0412143
\(718\) 4.61075 0.172072
\(719\) −16.2500 −0.606024 −0.303012 0.952987i \(-0.597992\pi\)
−0.303012 + 0.952987i \(0.597992\pi\)
\(720\) −3.77963 −0.140859
\(721\) −2.13919 −0.0796676
\(722\) 49.2537 1.83303
\(723\) −0.969052 −0.0360394
\(724\) 7.86910 0.292453
\(725\) 94.6444 3.51500
\(726\) −6.76589 −0.251106
\(727\) −14.0221 −0.520050 −0.260025 0.965602i \(-0.583731\pi\)
−0.260025 + 0.965602i \(0.583731\pi\)
\(728\) −2.13919 −0.0792837
\(729\) 1.00000 0.0370370
\(730\) −47.5603 −1.76029
\(731\) −32.7626 −1.21177
\(732\) 5.36119 0.198155
\(733\) −44.9814 −1.66142 −0.830712 0.556702i \(-0.812066\pi\)
−0.830712 + 0.556702i \(0.812066\pi\)
\(734\) 8.74574 0.322811
\(735\) −9.16132 −0.337920
\(736\) 0.374182 0.0137925
\(737\) 33.7801 1.24431
\(738\) −10.2607 −0.377701
\(739\) 26.3550 0.969486 0.484743 0.874657i \(-0.338913\pi\)
0.484743 + 0.874657i \(0.338913\pi\)
\(740\) 22.4255 0.824377
\(741\) 8.26158 0.303497
\(742\) −7.56016 −0.277542
\(743\) 25.8367 0.947857 0.473928 0.880563i \(-0.342836\pi\)
0.473928 + 0.880563i \(0.342836\pi\)
\(744\) 3.35956 0.123167
\(745\) −74.2628 −2.72078
\(746\) 17.1101 0.626447
\(747\) 4.49347 0.164407
\(748\) 19.6599 0.718837
\(749\) −6.18799 −0.226104
\(750\) 16.1981 0.591470
\(751\) 16.4576 0.600546 0.300273 0.953853i \(-0.402922\pi\)
0.300273 + 0.953853i \(0.402922\pi\)
\(752\) 5.37045 0.195840
\(753\) 14.0977 0.513747
\(754\) −10.1926 −0.371192
\(755\) −23.9736 −0.872490
\(756\) −2.13919 −0.0778016
\(757\) 10.3983 0.377931 0.188965 0.981984i \(-0.439487\pi\)
0.188965 + 0.981984i \(0.439487\pi\)
\(758\) 14.7409 0.535412
\(759\) −1.57716 −0.0572474
\(760\) −31.2257 −1.13268
\(761\) 50.6345 1.83550 0.917749 0.397161i \(-0.130005\pi\)
0.917749 + 0.397161i \(0.130005\pi\)
\(762\) 0.794271 0.0287734
\(763\) −27.2624 −0.986967
\(764\) 5.36373 0.194053
\(765\) −17.6294 −0.637392
\(766\) −3.02320 −0.109233
\(767\) −5.68060 −0.205115
\(768\) −1.00000 −0.0360844
\(769\) 5.75283 0.207453 0.103726 0.994606i \(-0.466923\pi\)
0.103726 + 0.994606i \(0.466923\pi\)
\(770\) −34.0794 −1.22814
\(771\) 27.4037 0.986920
\(772\) 20.9799 0.755085
\(773\) −34.4727 −1.23990 −0.619949 0.784642i \(-0.712847\pi\)
−0.619949 + 0.784642i \(0.712847\pi\)
\(774\) −7.02409 −0.252476
\(775\) −31.1956 −1.12058
\(776\) 10.7188 0.384783
\(777\) 12.6923 0.455335
\(778\) −28.4470 −1.01988
\(779\) −84.7694 −3.03718
\(780\) −3.77963 −0.135333
\(781\) −33.8778 −1.21224
\(782\) 1.74530 0.0624119
\(783\) −10.1926 −0.364253
\(784\) −2.42386 −0.0865666
\(785\) −37.4189 −1.33554
\(786\) 14.1470 0.504606
\(787\) 34.1799 1.21838 0.609192 0.793023i \(-0.291494\pi\)
0.609192 + 0.793023i \(0.291494\pi\)
\(788\) 5.11889 0.182353
\(789\) −19.1820 −0.682898
\(790\) −50.1935 −1.78580
\(791\) 8.05678 0.286466
\(792\) 4.21496 0.149772
\(793\) 5.36119 0.190381
\(794\) 21.7150 0.770638
\(795\) −13.3577 −0.473748
\(796\) 21.6419 0.767076
\(797\) −37.2281 −1.31869 −0.659344 0.751841i \(-0.729166\pi\)
−0.659344 + 0.751841i \(0.729166\pi\)
\(798\) −17.6731 −0.625621
\(799\) 25.0495 0.886187
\(800\) 9.28562 0.328296
\(801\) −11.6173 −0.410476
\(802\) 15.6403 0.552278
\(803\) 53.0382 1.87168
\(804\) −8.01434 −0.282644
\(805\) −3.02539 −0.106631
\(806\) 3.35956 0.118335
\(807\) 19.1913 0.675567
\(808\) −8.30659 −0.292225
\(809\) 55.6323 1.95593 0.977963 0.208778i \(-0.0669486\pi\)
0.977963 + 0.208778i \(0.0669486\pi\)
\(810\) −3.77963 −0.132803
\(811\) −37.5771 −1.31951 −0.659755 0.751480i \(-0.729340\pi\)
−0.659755 + 0.751480i \(0.729340\pi\)
\(812\) 21.8039 0.765165
\(813\) 21.9819 0.770939
\(814\) −25.0084 −0.876544
\(815\) −52.1880 −1.82807
\(816\) −4.66431 −0.163284
\(817\) −58.0301 −2.03021
\(818\) 11.2738 0.394177
\(819\) −2.13919 −0.0747494
\(820\) 38.7816 1.35431
\(821\) 2.66187 0.0928998 0.0464499 0.998921i \(-0.485209\pi\)
0.0464499 + 0.998921i \(0.485209\pi\)
\(822\) 11.3093 0.394456
\(823\) −2.92614 −0.101999 −0.0509995 0.998699i \(-0.516241\pi\)
−0.0509995 + 0.998699i \(0.516241\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −39.1385 −1.36263
\(826\) 12.1519 0.422818
\(827\) 10.1477 0.352869 0.176435 0.984312i \(-0.443544\pi\)
0.176435 + 0.984312i \(0.443544\pi\)
\(828\) 0.374182 0.0130037
\(829\) 17.1674 0.596250 0.298125 0.954527i \(-0.403639\pi\)
0.298125 + 0.954527i \(0.403639\pi\)
\(830\) −16.9837 −0.589511
\(831\) −14.0129 −0.486103
\(832\) −1.00000 −0.0346688
\(833\) −11.3057 −0.391718
\(834\) −20.2607 −0.701572
\(835\) −4.28287 −0.148215
\(836\) 34.8222 1.20435
\(837\) 3.35956 0.116123
\(838\) 6.51719 0.225133
\(839\) 17.0932 0.590124 0.295062 0.955478i \(-0.404660\pi\)
0.295062 + 0.955478i \(0.404660\pi\)
\(840\) 8.08535 0.278971
\(841\) 74.8886 2.58237
\(842\) 18.6962 0.644314
\(843\) 6.84656 0.235808
\(844\) 20.6224 0.709853
\(845\) −3.77963 −0.130023
\(846\) 5.37045 0.184640
\(847\) 14.4735 0.497316
\(848\) −3.53412 −0.121362
\(849\) −22.2633 −0.764074
\(850\) 43.3110 1.48556
\(851\) −2.22011 −0.0761045
\(852\) 8.03750 0.275360
\(853\) −21.3095 −0.729624 −0.364812 0.931081i \(-0.618867\pi\)
−0.364812 + 0.931081i \(0.618867\pi\)
\(854\) −11.4686 −0.392448
\(855\) −31.2257 −1.06790
\(856\) −2.89268 −0.0988697
\(857\) 25.7260 0.878784 0.439392 0.898295i \(-0.355194\pi\)
0.439392 + 0.898295i \(0.355194\pi\)
\(858\) 4.21496 0.143896
\(859\) 23.0583 0.786739 0.393370 0.919380i \(-0.371309\pi\)
0.393370 + 0.919380i \(0.371309\pi\)
\(860\) 26.5485 0.905295
\(861\) 21.9495 0.748038
\(862\) −18.5036 −0.630235
\(863\) 5.00615 0.170411 0.0852057 0.996363i \(-0.472845\pi\)
0.0852057 + 0.996363i \(0.472845\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 53.3866 1.81520
\(866\) −29.7200 −1.00993
\(867\) −4.75583 −0.161516
\(868\) −7.18674 −0.243934
\(869\) 55.9746 1.89881
\(870\) 38.5242 1.30609
\(871\) −8.01434 −0.271555
\(872\) −12.7443 −0.431576
\(873\) 10.7188 0.362777
\(874\) 3.09133 0.104566
\(875\) −34.6507 −1.17141
\(876\) −12.5833 −0.425151
\(877\) −8.41476 −0.284146 −0.142073 0.989856i \(-0.545377\pi\)
−0.142073 + 0.989856i \(0.545377\pi\)
\(878\) −19.7906 −0.667902
\(879\) −0.00371315 −0.000125242 0
\(880\) −15.9310 −0.537034
\(881\) −7.88171 −0.265542 −0.132771 0.991147i \(-0.542387\pi\)
−0.132771 + 0.991147i \(0.542387\pi\)
\(882\) −2.42386 −0.0816158
\(883\) 17.6173 0.592868 0.296434 0.955053i \(-0.404202\pi\)
0.296434 + 0.955053i \(0.404202\pi\)
\(884\) −4.66431 −0.156878
\(885\) 21.4706 0.721726
\(886\) 7.33689 0.246488
\(887\) 35.9708 1.20778 0.603890 0.797067i \(-0.293616\pi\)
0.603890 + 0.797067i \(0.293616\pi\)
\(888\) 5.93324 0.199107
\(889\) −1.69910 −0.0569859
\(890\) 43.9090 1.47183
\(891\) 4.21496 0.141206
\(892\) −6.51667 −0.218194
\(893\) 44.3684 1.48473
\(894\) −19.6482 −0.657133
\(895\) −70.7859 −2.36611
\(896\) 2.13919 0.0714653
\(897\) 0.374182 0.0124936
\(898\) 4.87953 0.162832
\(899\) −34.2426 −1.14205
\(900\) 9.28562 0.309521
\(901\) −16.4843 −0.549170
\(902\) −43.2483 −1.44001
\(903\) 15.0259 0.500029
\(904\) 3.76628 0.125265
\(905\) −29.7423 −0.988667
\(906\) −6.34285 −0.210727
\(907\) −33.0680 −1.09800 −0.549002 0.835821i \(-0.684992\pi\)
−0.549002 + 0.835821i \(0.684992\pi\)
\(908\) 0.198682 0.00659348
\(909\) −8.30659 −0.275512
\(910\) 8.08535 0.268027
\(911\) 28.4968 0.944142 0.472071 0.881561i \(-0.343507\pi\)
0.472071 + 0.881561i \(0.343507\pi\)
\(912\) −8.26158 −0.273568
\(913\) 18.9398 0.626815
\(914\) −20.0349 −0.662695
\(915\) −20.2633 −0.669885
\(916\) −26.2874 −0.868559
\(917\) −30.2631 −0.999375
\(918\) −4.66431 −0.153945
\(919\) −21.8077 −0.719371 −0.359686 0.933074i \(-0.617116\pi\)
−0.359686 + 0.933074i \(0.617116\pi\)
\(920\) −1.41427 −0.0466271
\(921\) −11.9442 −0.393575
\(922\) −18.6206 −0.613237
\(923\) 8.03750 0.264558
\(924\) −9.01660 −0.296624
\(925\) −55.0938 −1.81147
\(926\) 0.740569 0.0243366
\(927\) −1.00000 −0.0328443
\(928\) 10.1926 0.334588
\(929\) 20.8242 0.683220 0.341610 0.939842i \(-0.389028\pi\)
0.341610 + 0.939842i \(0.389028\pi\)
\(930\) −12.6979 −0.416381
\(931\) −20.0249 −0.656291
\(932\) −26.4007 −0.864784
\(933\) 4.77869 0.156447
\(934\) −13.1137 −0.429093
\(935\) −74.3072 −2.43010
\(936\) −1.00000 −0.0326860
\(937\) 21.2627 0.694621 0.347311 0.937750i \(-0.387095\pi\)
0.347311 + 0.937750i \(0.387095\pi\)
\(938\) 17.1442 0.559778
\(939\) 17.6126 0.574766
\(940\) −20.2983 −0.662059
\(941\) 54.3137 1.77058 0.885288 0.465043i \(-0.153961\pi\)
0.885288 + 0.465043i \(0.153961\pi\)
\(942\) −9.90015 −0.322564
\(943\) −3.83936 −0.125027
\(944\) 5.68060 0.184888
\(945\) 8.08535 0.263017
\(946\) −29.6063 −0.962582
\(947\) 10.7767 0.350194 0.175097 0.984551i \(-0.443976\pi\)
0.175097 + 0.984551i \(0.443976\pi\)
\(948\) −13.2800 −0.431314
\(949\) −12.5833 −0.408472
\(950\) 76.7139 2.48893
\(951\) 17.4337 0.565328
\(952\) 9.97786 0.323384
\(953\) −8.93686 −0.289493 −0.144747 0.989469i \(-0.546237\pi\)
−0.144747 + 0.989469i \(0.546237\pi\)
\(954\) −3.53412 −0.114421
\(955\) −20.2729 −0.656017
\(956\) −1.10359 −0.0356927
\(957\) −42.9613 −1.38874
\(958\) 18.4924 0.597462
\(959\) −24.1927 −0.781223
\(960\) 3.77963 0.121987
\(961\) −19.7134 −0.635915
\(962\) 5.93324 0.191295
\(963\) −2.89268 −0.0932152
\(964\) 0.969052 0.0312111
\(965\) −79.2965 −2.55264
\(966\) −0.800447 −0.0257539
\(967\) −22.8309 −0.734194 −0.367097 0.930183i \(-0.619648\pi\)
−0.367097 + 0.930183i \(0.619648\pi\)
\(968\) 6.76589 0.217464
\(969\) −38.5346 −1.23791
\(970\) −40.5132 −1.30080
\(971\) 55.4526 1.77956 0.889779 0.456392i \(-0.150858\pi\)
0.889779 + 0.456392i \(0.150858\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 43.3416 1.38947
\(974\) 32.5538 1.04309
\(975\) 9.28562 0.297378
\(976\) −5.36119 −0.171608
\(977\) 53.9815 1.72702 0.863511 0.504330i \(-0.168260\pi\)
0.863511 + 0.504330i \(0.168260\pi\)
\(978\) −13.8077 −0.441522
\(979\) −48.9663 −1.56497
\(980\) 9.16132 0.292648
\(981\) −12.7443 −0.406894
\(982\) 34.3094 1.09486
\(983\) 3.08431 0.0983742 0.0491871 0.998790i \(-0.484337\pi\)
0.0491871 + 0.998790i \(0.484337\pi\)
\(984\) 10.2607 0.327098
\(985\) −19.3475 −0.616463
\(986\) 47.5414 1.51403
\(987\) −11.4884 −0.365681
\(988\) −8.26158 −0.262836
\(989\) −2.62829 −0.0835747
\(990\) −15.9310 −0.506320
\(991\) −19.7165 −0.626316 −0.313158 0.949701i \(-0.601387\pi\)
−0.313158 + 0.949701i \(0.601387\pi\)
\(992\) −3.35956 −0.106666
\(993\) −11.6141 −0.368562
\(994\) −17.1937 −0.545353
\(995\) −81.7983 −2.59318
\(996\) −4.49347 −0.142381
\(997\) 30.8424 0.976789 0.488394 0.872623i \(-0.337583\pi\)
0.488394 + 0.872623i \(0.337583\pi\)
\(998\) −3.03217 −0.0959817
\(999\) 5.93324 0.187720
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8034.2.a.bc.1.2 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.2 15 1.1 even 1 trivial